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I was reading about star polygon graphs from the following link:

http://mathworld.wolfram.com/StarPolygon.html.

As far as I noticed I felt that whenever $d$ is a proper divisor of $n$, then we get $d$ copies of cycle of length $n/d$ where $~$ $d<\lfloor n\rfloor$. Is my observation correct? Kindly rectify me if I am wrong somewhere.

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Yes, your interpretation is consistent with the MathWorld entry.

However, it should be noted that not everyone uses this interpretation. In particular, Grünbaum and others (like me) take the model regular $\{n/d\}$-gon to have its $k$-th vertex (starting at the $0$-th) at coordinates $$\left(\;\cos \frac{2\pi dk}{n}\;,\; \sin\frac{2\pi dk}{n} \;\right)$$ With this view, a $\{12/4\}$-gon, for instance, isn't a compound of four separate triangles in the MathWorld sense; it's a dodecagon that wraps around a single triangular cycle four times. See some related thoughts in this answer to the question "What is a Hexagon?".

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